Many authors tried to study exact values of the generalized connectivity of graphs. The generalized \(k\) -connectivity of the complete graph, \(\kappa _k(K_n)\) , was determined in [1]} for every pair \(k,n\)  of integers with \(2\le k\le n\) . The generalized \(k\) -connectivity of the complete bipartite graphs \(K_{a,b}\)  are obtained in [2]} for all \(2\le k\le a+b\) . Apart from these two results, the generalized \(k\) -connectivity of other important classes of graphs, such as, Cartesian product graphs [3]}, [4]}, hypercubes [3]}, [6]}, several variations of hypercubes [7]}, [8]}, [9]}, [10]}, Cayley graphs [11]}, [12]}, [13]}, [14]}, have draw many scholars' attention. So far, as we can see, the results on the generalized \(k\) -connectivity of network are almost about \(k=3\) .
